Smooth and étale morphisms

Abstract: References: Mumford, The red book of varieties and schemes, III.5 and III.10; or Poonen, Rational points on varieties, Section 3.5.

Smooth varieties give an algebraic analogue of (smooth) manifolds from differential geometry, while smooth and étale morphisms give algebraic analogues of submersions and local isomorphisms. In addition to translating important notions from differential geometry into the algebraic setting, maps of these types play an important role in later development of étale cohomology. In this talk, we will introduce the definitions and basic properties of smooth and étale morphisms with an emphasis on providing intuition for thinking about them.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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